No, the Sobol index must be less than or equal to 1 :)

Hi Daniel,

Contrary to the previous reply, the Sobol' indices can actually sum to greater than one when you have correlated variables.

When input variables are independent, the variance of the model output can be decomposed into a nice neat decomposition. In this case, all the Sobol indices will sum to 1.

When you have correlated variables, this is not the case. An intuitive way to think about it is as follows: correlations mean that the information contained in a given variable is also partly contained in other variables with which it is correlated. Say you have a model with two variables, X1 and X2, where X1 and X2 are positively correlated. If you look at the sensitivity of the model to X1, you will get the sensitivity due to that variable, plus a "bit" of sensitivity due X2, because by changing X1 you are also influencing X2. If you now look at the sensitivity due to X2, you will get the sensitivity due to X2 on its own, plus a "bit" of the sensitivity of X1. Now, if you add up these sensitivity indices, they will be greater than one because you count the "bits" twice: once with X1 and once with X2. The greater the correlation, you will get a sum that is increasingly greater than 1.

So the variance decomposition for correlated variables is more complicated, because you have chunks of variance due to correlations.

A fairly accessible paper that explains this in the introduction is here: https://core.ac.uk/download/pdf/81969682.pdf But there are plenty other papers around if you search for "correlated sensitivity analysis" in Google Scholar. I don't know whether or not you have access to journals so the paper I linked here is open-access.

Cheers

Will

The paper "A new framework of variance based global sensitivity analysis for models with correlated inputs" may be helpful for your question

Hi guys,

Thanks a lot! This will help tons in understanding.

Daniel

Hi? For the Sobol Analysis, you might want to calculate the second-order indices to examine how the interaction between a pair of inputs influence the model output. If there are no higher-order interactions (more than order 2), doing a few arithmetic operations may help you get the sole contributions of the input factors as 1. Have a look at this, might be helpful. Article A cloud-based framework for sensitivity analysis of natural ...